wrong and one correct, may not be distinguishable goodness of fit criteria, yet one may predict far ter parameter values than the other. In the absence other information, such as firm knowledge of the rect model or independent determinations of some the parameter values, one will evidently always nd an appreciable chance of picking a "best" model ich yields some quite poor parameter estimates. e better the accuracy of the data and the wider its ge, the better the higher-order parameter estimates l be since the final model chosen will be forced to closer to the correct model to achieve an adequate

The monotonic increase of d¡ and Sd with fitting range istrated in figures 6 to 12 is, of course, indicative of e use of an incorrect and inadequate model and is no means limited to the equation of state area. In >st if not all cases of practical interest, we may expect find this sort of behavior: the wider the range used least-squares fitting of a possible, "close," but still correct model, the greater will be sa and the paramer error magnitudes. It should, however, be remarked at this conclusion only applies in the usual case here the model is not asymptotically correct as the nge is extended indefinitely. The wider the range ed, generally the more difficult it will be for an correct model to simulate the correct one.

This increase of errors with range may frequently be ed in practical cases as a powerful means of disiminating against incorrect models. When random rors in the data are sufficiently small that the sysmatic errors arising from an incorrect model choice ominate sa, it will generally increase with the fitting nge, as illustrated here. Such an increase thus clearly gnals an incorrect model choice for the range of data ted. Since most models only apply adequately in any ase over a limited range of a variable such as pressure temperature, extension of the fitting range beyond e region of applicability of the best available model ill always eventually result in an increase in sa. Thus, any least squares fitting where the range of applicality of the model used in unknown, extrapolation tside the fitted range of data should be approached ith the utmost caution and avoided if possible. The present paper deals with exact data and actual -lative errors of parameters, but true parameter errors ill not be available in a usual experimental situation. evertheless, when sa increases because of the wrong odel choice, the estimated parameter standard deations will generally increase for the same reason. hus, these quantities, ordinary results of a least quares fitting, may also be used along with sa to help scriminate against an inadequate model.

There are some interesting general aspects to the resent results obtained with least-squares fitting of e wrong model. The residuals (here given by observed alues minus calculated values) show the following ehavior: The number of runs (number of sign changes lus one), u, for the ME1, BE1, and UTE, for which =3, is 4, while u=5 for the remaining equations for hich n=4. The general result, u= (n+1), is not very urprising but bears emphasizing. Further, the sign of he first residual run (which, together with knowledge

of u determines the signs and order of all the runs) is specific to the equation considered. For the present fitting of 3DGE data, this sign is +, −, −, +, −, −, + for the ME1, BE1, UTE, ME2, KE, BE2, and 3SE, respectively. The number of runs and their sign distributions were invariant in the present situation to the following: (a) p or V weighting, (b) the range of the data and its placement (all low p, all high p, all in the middle, etc.), and (c) the sign of V. Even though not all extremes were investigated, this high degree of pattern invariance is likely to be quite general and may itself be of considerable usefulness in helping to distinguish models.

Although we have not done it, one could readily establish a matrix of first signs obtained using data calculated from one of the present eight specific equations and fitted with another one of the eight. Then, in practical situations where it was believed that the correct model was one of the eight, many possibilities could be quickly eliminated by comparison with the sign of the first residual run obtained on fitting the actual data. This would only work, of course, provided random errors were considerably smaller than systematic ones and hence didn't appreciably perturb the residual pattern. With many data points, considerable perturbation of this kind could be tolerated, however, since decisions could be made on the basis of a smoothed residual pattern rather than the actual noisy pattern.

A partial comparison of the above type has been made earlier for the ME, and UTE [14]. There, Vo was taken fixed, so n=2. As expected, u was found to be three for both UTE fitting of exact ME, data and for ME, fitting of UTE data. The initial run signs were +, -, respectively, for the above two fittings.

5. Summary

This paper has been primarily concerned with discussing methods of discriminating between specific equations of state and has demonstrated considerable limitations on the possibility of adequate discrimination between "close" equations. We have found the somewhat surprising result that equations which cannot be adequately discriminated on the basis of least squares goodness of fit over even a wide pressure range may yet lead to estimates of higher-order parameter relative errors differing in sign and by an order of magnitude in absolute value for even a narrow pressure range, much less a wide one. The present methods, results, and conclusions can be generalized to a considerable degree to apply to model discrimination outside the equation of state area and are pertinent for linear models and for those nonlinear in their parameters, independent variable, or both. Thus, the following general conclusions, based on the present specific results, are likely to apply widely to the general data analysis field.

More than one mathematical model should usually be tested against the data in order to select, if possible, that model which fits best by objective criteria. As the range of data is progressively increased for which least squares fittings are carried out, the initial or eventual appearance of increases in sa and (sa), indicates the

FIGURE 13. General block diagram for data analysis. presence of systematic fitting error arising from an inadequate model choice. Such error will also usually show up in highly correlated residuals exhibiting, at least approximately, a number of sign-changes equal to the number of fitted parameters. The range of a causal experimental variable such as pressure, voltage, temperature, etc. should be increased to the maximum degree possible in order to allow the testing of a model for adequacy over the widest practical data range.

When two or more models have been found that represent the data over a given range with approximately the same goodness of fit and without signs of systematic errors from wrong model choice, it is still possible that one or more models may yield much better or much worse least-squares parameter estimates than the others. Additional independent information about likely parameter value ranges will usually then be necessary to allow a selection of the most appropriate equation for parameter estimation. Extrapolation of a given model-parameter value set beyond the range of data on which it is based is always dangerous.

When data smoothing or interpolation is the object, the possibility of discrimination between two models which yield equally good least squares fits to the data should be examined by the case B procedure of section 3. If the differences in dependent variables calculated with the same reasonable set of parameter values in each model are comparable to or smaller than the estimated random errors in the data, discrimination is impractical for that data set.

Figure 13 shows, in very diagrammatic form, appropriate steps in data analysis aimed at establishing a "best" model (including specific parameter value estimates). Some of the actually interrelated steps involved in this figure are presented differently in the flow chart of figure 14. This diagram is included for the benefit of those readers who may wish to apply the procedures discussed in this paper to other discrimination and parameter estimation problems.

For figure 14 we have assumed that a data set over a range, Rmax, has been taken, and that we wish to test potential models over the maximum range if possible.

The flow chart orders the tests as (1) case B, (2) runs. and (3) case A. If no models appear appropriate after the first series of tests, provision is made for decreas ing the range of the data used in the tests in order to determine the acceptable maximum range for parameter estimation.

In the flow chart, we have abbreviated the test for case B by the notation |▲yij|<σy. Here we mean that all or nearly all of the deviations should be less than the expected errors in the data. Note that "nearly all" is appropriate because of the possible presence of random outliers. For the same reason, the test u>n+2 should also be considered approximate and applied judiciously. Note also that σy may vary with x (het eroscedastic case); the test should be so applied when appropriate. Other symbols introduced here are € defined to be the acceptable level for standard devia tion of the least squares fitting, and ea, defined as the level below which standard deviations of two separate fits are indistinguishable.

Good data are usually expensive, yet too little adequate data analysis is the general rule. It is better to do too much such analysis than too little.

We much appreciate the very helpful comments of E. L. Jones and of a referee.

6. References

[1] Macdonald, J. R., Review of some experimental and analytical equations of state, Rev. Mod. Phys. 41,316 (1969).

[2] Davis, L. A., and Gordon, R. B., Compression of mercury at high pressure, J. Chem. Phys. 46, 2650 (1967).

[3] Barsch, G. R., and Chang, Z. P., Ultrasonic and static equation of state for cesium halides, Proc. Symp. on Accurate Characterization of the High Pressure Environment, National Bureau of Standards, Gaithersburg, Md., Oct. 14–18, 1968. (NBS Spec. Publ. No. 326, pp. 173-186). The room-temperature Csl parameter values used in the present work were taken from a preprint of this paper distributed by the authors. Values appearing in the final version are Ko=118.9±0.5 kbar, K=5.93±0.08, and K-0.073±0.008 kbar-1. Note that this │Kő| is nearly three of its standard deviations larger than the value of -0.052±0.002 kbar-1 presented earlier.

[4] Lazarus, D., The variation of the adiabatic elastic constants of KCl, NaCl, CuZn, Cu, and Al with pressure to 10,000 bars, Phys. Rev. 76, 545 (1949).

[5] Ruoff, A. L., Linear shock-velocity-particle-velocity relationship, J. Appl. Phys. 38, 4976 (1967).

[6] Monfort III, C. E., and Swenson, C. A., An experimental equation of state for potassium metal, J. Phys. Chem. Solids 26, 291 (1965).

[7] Kell, G. S., and Whalley, E., The PVT properties of water. I. Liquid water in the temperature range 0 to 150 °C and at pressures up to 1 kbar, Phil. Trans. Roy. Soc. (London) A258,565 (1965).

[8] Vedam, R., and Holton, G., Specific volumes of water at high pressures obtained from ultrasonic-propagation measurements, J. Acoust. Soc. Am. 43, 108 (1968).

[9] Macdonald, J. R., Accelerated convergence, divergence, iteration, extrapolation, and curve fitting, J. Appl. Phys. 35, 3034 (1964).

[10] Wampler, R. H., A report on the accuracy of some widely used least squares computer programs, J. Am. Stat. Assoc. 65, 549 (1970).

[11]

[12]

Deming, W. E., Statistical Adjustment of Data (John Wiley &
Sons, Inc., New York, 1943).

O'Neill, M., Sinclair, I. G., and Smith, F. J., Polynomial curve
fitting when abscissas and ordinates are both subject to error,
Computer J. 12, 52 (1969).

[13] Powell, D. R., and Macdonald, J. R., A rapidly convergent iterative method for the solution of the generalized nonlinear least squares problem, submitted to Computer J.

[14] Macdonald, J. R., Some simple isothermal equations of state, Rev. Mod. Phys. 38, 669 (1966).

(Paper 75A5-678)

1. Introduction

In this paper we report rotational analyses of several mportant new bands observed in the absorption specrum of carbon monoxide. These analyses extend the observed data for four excited electronic states, a' 3Σ+, e 3Σ−, I 1Σ-, and D 1A, which have previously been characterized in the literature.

Herzberg and Hugo [1] first characterized the forbidden a' -X and e --X' absorption systems of CO. For the a' -X system they analyzed X system they analyzed hirteen bands ranging from v' = 2 to 23 of the a' state. Krupenie [2] has pointed out the need for further experimental work to determine whether the upper state of the two emission bands at 2670 and 2980 Å, tentatively associated with the fall transition, should be interpreted in terms of low vibrational levels of a separate electronic state, f3Σ+, or interpreted in terms of high vibrational levels, namely v=31 and 35 of the a' 3 state as originally proposed by Stepanov 3]. One objective in this work was to try to extend the observations of the a' state to higher vibrational levels beyond the v=23 level observed earlier. Although we observed five new a'-X bands terminating on low vibrational levels of the a' state, only one new band was observed terminating on higher vibrational levels. The latter band at 87428.4 cm-1 has been assigned as a'-X (38-0). Based on this assignment, arguments will be made in favor of Stepanov's assignment of the emission bands to the a' 3-a 3II transition. Herzberg and Hugo [1] analyzed six bands of the e-X system. ranging from v' = 2 to 10. We have analyzed eleven additional bands of this system that extend the obser

*E. O. Hulburt Center for Space Research, Naval Research Laboratory, Washington, D.C. 20390.

Figures in brackets indicate the literature references at the end of this paper.

vations to v' = 27. One of the new bands was on the long wavelength side of the bands observed earlier, indicating that the vibrational numbering of the e state should be increased by one unit. All statements concerning the e state in this paper have taken that numbering change into account.

The new vibrational numbering of the e state and the original numbering of the a' state were both confirmed by the analyses of 13C16O isotopic bands.

Previous papers have characterized the forbidden. '--XΣ+ [4] and D'A-X + [5] absorption systems. We report here the analyses of two new bands of the I-X system and one new band of the D-X systems. These new bands involve transitions to high vibrational levels of the I and D states.

Revised vibrational and rotational coefficients and equilibrium constants have been determined for the a', e, I, and D states by combining band origins and B-values from the earlier studies with similar data from the new analyses given here.

2. Experimental Procedure

Two sets of high-resolution absorption plates were used in the analyses. One set was taken at the National Research Council, Ottawa, Canada. The region 1550 to 1220 A was photographed in the seventh, eighth, and ninth orders of the 10-m vacuum spectrograph described by Douglas and Potter [6]. The continuum light source was a Lyman discharge through argon. Overlapping orders were separated by a lithium fluoride prism-cylindrical lens combination described by Brix and Herzberg [7].

The second set of plates was taken at the U.S. Naval Research Laboratory. The region 1800 to 1060 A was photographed in the third, fourth, and fifth orders of a